MATLAB, DSP, Julia, Power quality, Engineering, Metrology

Calculate Powers in Non-sinusoidal Conditions According to IEEE 1459

Feb 25th, 2015

Definitions of electric power qualities under nonsinusoidal conditions are quite complex and hard to grasp.
Annex B of the IEEE Std 1459-2010 shows how to calculate apparent power components in non-sinusoidal conditions. I have written a MATLAB script to illustrate this example. Unfortunately, there were typo errors in the standard, that’s why some of the values are different in my example.

It is learned from this expression that every component of $ S $ contributes to the total power loss in the supplying system. This means that not only fundamental active and reactive powers cause losses but also the current and voltage distortion powers as well as the harmonic apparent power cause losses.

The following numerical example is meant to facilitate the understanding of the previous explanations:

The total harmonic active power $ P_H =P –27.47 W < 0 $ is supplied by the load and injected into the power system. This condition is typical for dominant nonlinear loads. The bulk of the active power is supplied to the load by the fundamental component $ P_1 = 8660.25 W$ .

Of interest is the fact that $ Q_5 < 0$ , whereas other reactive powers are positive. If one incorrectly defines a total reactive power as the sum of the four reactive powers (in accordance with C. Budeanu’s definition):

\[
Q_B = Q_1 +Q_3 +Q_5 +Q_7 = 4984.67 \, var
\]

and assumes that the supplying line has a resistance $ r= 1.0 \, \Omega $ and the load is supplied with an rms voltage $ V=240 V $, the power loss due to $ Q_B $ in line is

The reactive power $ Q_5 $ , despite its negative value, contributes to the line losses in the same way as the positive reactive powers. The fact that harmonic reactive powers of different orders oscillate with different frequencies reinforces the conclusion that the reactive powers should not be added arithmetically (as recommended by Budeanu).

The cross-products that produce the distortion powers $ D_I $ and $ D_V $ are given in Table 3.

The apparent power and its components are represented in the following tree:

Figure 2 — Apparent power and its components tree

The fundamental power factor (displacement power factor) is $ PF_1 = P_1 / S_1 =0.866 $, and the power factor is $ PF= P/S=0.821 $ . The dominant power components are $ P_1 $ and $ Q_1 $ . Due to relatively large distortion, $ S_N $ is found to be a significant portion of $ S $, and the current distortion power $ D_I $ is the dominant component of $ S_N $ .